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Metamath Proof Explorer

Theorem brric

Description: The relation "is isomorphic to" for (unital) rings. (Contributed by AV, 24-Dec-2019)

Ref Expression
Assertion brric ( 𝑅 ≃𝑟 𝑆 ↔ ( 𝑅 RingIso 𝑆 ) ≠ ∅ )

Proof

Step Hyp Ref Expression
1 df-ric ⊢ ≃𝑟 = ( ◡ RingIso “ ( V ∖ 1o ) )
2 ovex ⊢ ( 𝑟 RingHom 𝑠 ) ∈ V
3 rabexg ⊢ ( ( 𝑟 RingHom 𝑠 ) ∈ V → { ℎ ∈ ( 𝑟 RingHom 𝑠 ) ∣ ◡ ℎ ∈ ( 𝑠 RingHom 𝑟 ) } ∈ V )
4 2 3 mp1i ⊢ ( ( 𝑟 ∈ V ∧ 𝑠 ∈ V ) → { ℎ ∈ ( 𝑟 RingHom 𝑠 ) ∣ ◡ ℎ ∈ ( 𝑠 RingHom 𝑟 ) } ∈ V )
5 4 rgen2 ⊢ ∀ 𝑟 ∈ V ∀ 𝑠 ∈ V { ℎ ∈ ( 𝑟 RingHom 𝑠 ) ∣ ◡ ℎ ∈ ( 𝑠 RingHom 𝑟 ) } ∈ V
6 df-rim ⊢ RingIso = ( 𝑟 ∈ V , 𝑠 ∈ V ↦ { ℎ ∈ ( 𝑟 RingHom 𝑠 ) ∣ ◡ ℎ ∈ ( 𝑠 RingHom 𝑟 ) } )
7 6 fnmpo ⊢ ( ∀ 𝑟 ∈ V ∀ 𝑠 ∈ V { ℎ ∈ ( 𝑟 RingHom 𝑠 ) ∣ ◡ ℎ ∈ ( 𝑠 RingHom 𝑟 ) } ∈ V → RingIso Fn ( V × V ) )
8 5 7 ax-mp ⊢ RingIso Fn ( V × V )
9 1 8 brwitnlem ⊢ ( 𝑅 ≃𝑟 𝑆 ↔ ( 𝑅 RingIso 𝑆 ) ≠ ∅ )